Here we study a generalization of Cannings'
models that, nonetheless, still fall under
the umbrella of neutral evolution. Our results provide weaker conditions for the main result in [1]}
(see Theorem REF  and Proposition REF ), expanding its applicability and providing a better insight into neutral evolution from a mathematical perspective. Having an accurate understanding of the neutral case is paramount for the theory and applications where the effect of natural selection is studied. One of our main contributions is the change in
focus in neutral models from the exchangeability (symmetry) condition on
offspring distributions, to the weaker condition of non-heritability of reproductive success.
It is in fact from the latter property, and not the former,
from which neutrality stems; both, from biological
(heritability is a necessary condition for natural selection to occur, whereas asymmetric reproductive success is not sufficient) and mathematical perspectives (see the discussion below).
